Streaming Potential with Ideally Polarizable Electron-Conducting Substrates

With nonconducting substrates, streaming potential in sufficiently broad (vs Debye screening length) capillaries is well known to be a linear function of applied pressure (and coordinate along the capillary). This study for the first time explores streaming potential with ideally polarizable electron-conducting substrates and shows it to be a nonlinear function of both coordinate and applied pressure. Experimental manifestations can be primarily expected for streaming potentials arising along thin porous electron-conducting films experiencing solvent evaporation from the film side surface. Model predictions are in good qualitative agreement with literature experimental data.


■ INTRODUCTION
Foundations of the classical theory of streaming potential were laid down more than a century ago. 1 The initial and consequent models have considered non-electron-conducting substrates. 2 Several studies considered ion-conducting, namely, porous substrates (see, for example, refs 3−5) but the physics in this case is essentially different due to the relatively low conductivity of such substrates and lack of ideal polarizability of interfaces between them and electrolyte solutions. Recently, a new interesting context for electrokinetic phenomena of streaming potential (and streaming current) has arisen in capillarity-driven energy harvesting from evaporation with (nano)porous materials (see, for example, refs 6−9; the state of the art of this emerging field has very recently been critically reviewed in ref 10). In this case, hydrostatic pressure drops can be very high being ultimately controlled by capillary pressures in nanopores. At the same time, several relevant experimental studies used electron-/hole-conducting nanoporous substrates. 8,9,11 Below, we will see that the combination of large hydrostatic pressure drops with solid-substrate electron conductance can make steaming potential essentially different from the classical case.
Some transport phenomena (membrane potential, electrical conductance, and pressure-driven salt rejection) in electrolytefilled nanopores with electron-conducting walls have been recently explored by Ryzhkov et al. 12−17 These phenomena are nontrivial only when there is a noticeable overlap of diffuse parts of electric double layers (EDLs). Such systems afford only numerical analysis. Besides, principal emphasis was made on the impact of an external bias while the role of redistribution of electron charges in floating (ungrounded) systems was less explored. Electrokinetic phenomena were not considered.
In this study, for the first time, we account for electron/hole conductance of matrices of porous materials experiencing flowinduced streaming potential. To obtain simple analytical results, we consider the limiting case of sufficiently broad capillaries without any appreciable overlap of diffuse parts of EDLs and surface conductance phenomena as well as neglect the existence of the so-called Stern layer. 18 The existence of the latter may be important in more concentrated solutions and close to strongly charged surfaces. Below, we will see that strong surface charges (chemical plus induced) may well arise close to the channel exit under strongly nonlinear conditions. Therefore, accounting for the Stern layer is an essential next step that will be made in future studies.
We also discuss scenarios of possible experimental manifestations and demonstrate that they can be expected for rather large hydrostatic pressure differences and in sufficiently dilute electrolyte solutions. In combination with the requirement of negligible EDL overlap (implying relatively large pore size), this may be difficult to achieve in pressuredriven processes. However, we will see that situation can be different in systems where large hydrostatic pressure gradients are induced by capillarity in water evaporation from hydrophilic nanopores.

■ THEORY
Streaming currents arise as a result of the advective movement of electrically charged liquids close to "charged" solid/liquid interfaces in electrolyte solutions. Strictly speaking, the total electric charge of the interface region is zero; however, a charge is "bound" to the surface while its "counter-charge" can move with and/or relative to the liquid. Advective movement of electrolyte solution through a capillary with "charged" walls gives rise to a convective current. In streaming potential mode, external circuit is open, so the net electric current must be zero in any capillary cross section. Streaming potential is the voltage arising to compensate exactly the convective streaming current by an electromigration current in the opposite direction. The local density of convective current is equal to the product of local electric charge density and fluid velocity. Expressing the space-charge distribution via electrostatic potential by the Poisson equation, using the Stokes equation with the standard boundary condition of no slip on the capillary wall and taking into account the zero-current condition, for sufficiently broad (compared to the Debye screening length) capillaries, one can obtain this celebrated Smoluchowski formula 2 where φ is the electrostatic potential in the central part of the capillary (far away from its walls), ζ is the potential drop within diffuse parts of EDLs occurring at the surface (the so-called ζpotential), εε 0 is the fluid dielectric constant, η is the fluid viscosity, g is the (bulk) electrical conductivity of electrolyte solution. The potentials and the system of coordinate are schematically shown in Figure 1. With electron-conducting substrates, the electrostatic potential of the conductor surface must be the same all the way along the capillary. Let us denote its constant value by Φ + = By substituting eq 2 into eq 1, we obtain In this simple analysis, we neglect the influence of electrokinetic phenomena on the volume flow (this is justified in sufficiently broad capillaries). Therefore, the hydrostatic pressure is independent of electrostatic potential and its profile is linear. Accordingly, eq 3 can be easily integrated along the capillary to yield where L is the channel length is the hydrostatic-pressure difference along the capillary. From eqs 4 and 5, we obtain The hydrostatic pressure profile is linear where ξ ≡ x/L is the dimensionless coordinate along the channel. Taking this into account, from eq 4, we obtain In contrast to the classical case of dielectric substrates, the electrostatic potential profile is nonlinear. The extent of nonlinearity is controlled by parameter A.
This analysis assumes that there are some fixed charges on the capillary walls (sometimes referred to as "chemical charge") arising due to preferential ion adsorption or dissociation of ionogenic groups. Therefore, there is a nonzero ζ-potential at zero volume flow. Under flow conditions, additional electrostatic potential arises outside EDLs and at the capillary walls owing to the appearance of net electric charges at the capillary edges. Physically, the constancy of surface electrostatic potential is ensured by the appearance of polarization electron/hole charges at the capillary surface. Together with the initially present "chemical" charges, these polarization charges give rise to a position-dependent ζpotential that can be found from the condition of constancy of full surface electrostatic potential (eq 2) and distribution of electrostatic potential outside the EDLs (eq 9) We consider the conductor ungrounded. Therefore, the total induced electron/hole charge must be zero. There is this wellknown relationship between surface-charge density and equilibrium electrostatic potential at a charged surface 2 so surface-charge density is proportional to the hyperbolic sinus of ζ-potential. For simplicity, let us initially assume that the "chemical" charge remains unchanged under flow conditions (constant charge approximation). Taking into account this and the fact that the total surface charge under flow conditions must be equal to the "chemical" charge, we obtain where ·

( )
RT c 2 2 sinh 0 0 2 0 is the (coordinate-independent) density of "chemical" surface charge and ζ 0 is the ζ- can be rewritten as Substituting eq 10 for the distribution of ζ-potential, we obtain The integral in the right-hand side of eq 14 can be taken to yield where Shi is the integral hyperbolic sinus. Equation 15 is a transcendental equation for the determination of (Φ − φ(0)) as a function of parameter A, which is proportional to the hydrostatic pressure drop along the channel. Φ and φ(0) enter eq 15 only in combination, (Φ − φ(0)), so they cannot be determined separately. However, the pressure dependence of streaming potential (eq 6) and distribution of ζ-potential (eq 10) depend only on this combination. When parameter A is small, by developing eq 15 in Taylor series in A, we can see that Φ − φ(0) ≈ ζ 0 . By substituting this into eq 9 and developing the exponential function in series for small A, we obtain which is the same behavior as in the classical case of dielectric substrates. Above we have considered the simplest case of constant "chemical" charge density independent of ζ-potential. Given that this charge is a result of preferential ion adsorption or dissociation of ionogenic groups, it typically depends on the concentration of some ions at the surface. This, in turn, is affected by electrostatic attraction/repulsion. Therefore, generally, the density of "chemical" surface charge should be considered a function of ζ-potential, σ(ζ). In the case of electron-conducting substrates, this potential is controlled not only by the "chemical" charge but also by the electron/hole polarization charges. However, whatever the mechanism of "chemical"-charge formation, the total polarization charge must be zero for ungrounded conductors. Therefore, the right-hand side of eq 12 should still be equal to the total "chemical" charge. With a charge regulation, the latter becomes dependent on ζ-potential, which changes with coordinate according to eq 10. Hence, on the left-hand side of eq 12, we should average the "chemical" charge density over the capillary length to obtain For the distribution of ζ-potential, we can still use eq 10, so where we have taken the integral from the right-hand side of eq 17. As previously, (Φ − φ(0)) can be found from eq 18 solved as a transcendental equation.
Within the scope of the popular charge regulation model, 19 the surface charge is described by the so-called Langmuir− Stern isotherm, which gives where Z p is the charge (in proton-charge units) of potentialdetermining ions and σ 0 is the maximum surface-charge density corresponding to full dissociation. Constant K is proportional to the bulk concentration of potential-determining ions and, thus, is a function of solution pH, for example, in the case of weakly acidic groups. The term with the exponent in the denominator reflects the fact that the surface concentration of ions is different from their bulk concentration due to electrostatic repulsion/attraction. Thus, for instance, an increase in the negative surface-charge density with increasing pH is accompanied by the intensification of electrostatic attraction of H + ions, which somewhat reduces the degree of dissociation.
Especially large values of dimensionless pressure differences can be expected in capillarity-driven electrokinetic phenomena, in particular, in systems with side evaporation from thin (nano)porous films (see Figure 4 for the schematic). A simple model for the distribution of hydrostatic pressure in such systems has recently been developed in ref 10 using the Darcy law for the description of viscous flow along the film and assuming a constant evaporation rate (controlled by the external mass transfer) from the fully wet part of the film. Under these assumptions, one obtains a linearly decreasing hydrostatic-pressure gradient along the film (in contrast to the constant pressure gradient occurring in the pressure-driven mode), which is because ever more liquid is lost to evaporation while moving along the film. The corresponding expression is where P is the hydrostatic pressure, x is the coordinate along the film, h is the film thickness, L is its length, χ is its hydraulic permeability, and q e is the linear evaporation rate (m/s). The evaporation rate is assumed to be constant along the film (we disregard the dependence of saturated-vapor pressure on the menisci curvature). After integration along the film (taking into account that pressure at the immersed end equals atmospheric (zero relative) pressure), we obtain As discussed in ref 10, the tangential hydraulic flow is driven by the gradient of (negative) capillary pressure arising beneath the curved menisci at the external film surface. While moving Langmuir pubs.acs.org/Langmuir Article along the film away from the immersed end, ever larger negative pressures are required to drive the viscous flow along the ever longer film segment. This negative-pressure buildup occurs due to a gradually increasing menisci curvature, which keeps growing until it reaches the maximum corresponding to the pore size. Once this state is reached, the menisci start to recede into the pores. Thus, the maximum pressure difference along the film is equal to the maximum negative capillary pressure. The length of the fully wet zone can be found by substituting negative maximum capillary pressure, −P cm , into eq 21 For the gradient of streaming potential, eq 1 is still applicable, though the hydrostatic-pressure gradient is not constant anymore but is given by eq 20 from which we obtain After integration where ξ ≡ x/L is the dimensionless coordinate along the porous film Taking into account as previously that the total induced electron/hole charge is zero and using eq 11, in the approximation of constant "chemical" charge, we obtain where ∑ is the surface tension, θ is the contact angle, and r p is the pore radius.

■ RESULTS AND DISCUSSION
Taking into account that integral hyperbolic sinus is a strongly increasing function of its argument, eq 15 shows that when parameter A increases, (Φ − φ(0)) → 0. Physically, this means that the polarization charges distribute in such a way that the net surface-charge density (fixed plus induced charges) at the capillary "entrance" tends to zero, whereas it "peaks" exponentially ever stronger (with increasing pressure difference) close to the "exit" (see eq 10). This is illustrated in Figure 2.    Figure 3 confirms the good applicability of eq 31. Figure 4 shows a comparison of pressure dependence of streaming potential calculated for the case of charge regulation using eqs 6 and 18 with the case of constant charge (eqs 6 and 15). The maximum surface-charge density in the case of charge regulation, σ 0 , is assumed to correspond to the same "zeroflow" dimensionless ζ-potential as in the case of constant charge.
As we can see, charge regulation can make the nonlinearity occur at somewhat larger dimensionless hydrostatic pressure differences, but qualitatively the behavior remains the same.
Scenarios of Experimental Verification. Pressure-Driven Mode. Above, we have seen that the extent of the nonlinearity (which distinguishes SP with electron-conducting substrates from the classical case) is directly proportional to the hydrostatic pressure drop and inversely proportional to the solution conductivity. Using the model of identical straight parallel cylindrical capillaries, the pressure drop can be expressed this way where J v is the volume flux (m/s), r p is the pore radius, and γ is the porosity (for tortuous pores, it also includes a tortuosity factor). Accordingly (see eq 7) Figure 2 shows that the nonlinearity becomes noticeable when A ≥ 2 ÷ 3. This parameter gets larger, in particular, in solutions of lower electric conductivity. For our simple model to be applicable, the capillaries have to be sufficiently broad compared to the thickness of diffuse parts of EDLs. The latter is known to increase with decreasing electrolyte concentration (solution conductivity) inversely proportionally to the square root of it. 20 Therefore, to maintain the impact of diffuse parts of EDLs at an acceptably low level, a decrease in concentration should be accompanied by an increase in the capillary radius. The latter implies less pressure drop at a given volume flux. As we can see from eq 33, parameter A is inversely proportional to the square of capillary radius. Therefore, at a given volume flux, reducing electrolyte concentration (and proportionally increasing the capillary radius) would leave parameter A (and the extent of nonlinearity) unchanged. At the same time, this would lead to an increase in Reynolds number and (in sufficiently broad capillaries) may result in deviations from the laminar flow pattern. 21 Another way to increase the "effective pressure drop" is using thicker diaphragms with relatively small pores. Thus, for instance, assuming the thickness (capillary length) of L = 1 cm, the capillary radius of r p = 0.5 μm, the active porosity of γ = 0.1, 1 mM NaCl solution, and a "reasonable" linear filtration rate of 0.3 mm/s, we obtain A ≈ 7. In 1 mM electrolyte solutions of (1:1) electrolytes, the EDL thickness is about 10 nm, which is around 50 times less than the assumed capillary radius (hence, no EDL overlap and surface conductance). Therefore, to achieve this flow rate in such a diaphragm, a pressure difference of about 1 MPa has to be applied. Sintered metals with average pore sizes down to single micrometers are commercially available. 22 Nevertheless, exploration of pronounced nonlinearity requires other recipes.
Evaporation-Driven Mode. In hydrophilic nanopores, capillary pressures can be very high (>10 MPa). In this subsection, we will demonstrate that this can lead to very large pressure differences along thin nanoporous films under evaporation conditions. These, in turn, can give rise to large "dimensionless pressures". In some studies, thin nanoporous films were assembled from electron-conducting nanoparticles (for example, carbon black). 7−9 In a typical configuration (see Figure 5), a thin (supported or stand-alone) film of a nanoporous material is immersed with one extremity in an   Assuming as previously a 1 mM aqueous NaCl solution, the pore radius of 0.5 μm, and perfect wetting according to eq 29 (θ = 0), for the dimensionless parameter B occurring at the maximum wet length, we obtain B m ≈ 4. In contrast to parameter A (see eq 33) controlling the nonlinearity in the pressure-driven mode, parameter B is inversely proportional to the first power of pore radius. Therefore, reducing electrolyte concentration (and conductivity) while increasing the pore size to keep the ratio of pore radius and screening length constant causes an increase in parameter B m inversely proportional to the square root of concentration. Thus, with 0.01 mM NaCl solution (and 5 μm pore radius), B m ≈ 40. Incidentally, experimental studies used very dilute electrolyte solutions although the pore size was essentially smaller than 10 μm. Figure 5 shows examples of distribution of electrostatic potential derivative along porous film with evaporation. This distribution is in good qualitative agreement with experimental data obtained in ref 8 for nanoporous films made from carbon black nanoparticles (see Figure 2 of ref 8) (Figure 6).

■ CONCLUSIONS
In the limiting case of sufficiently broad capillaries ("Smoluchowski limit"), streaming potential has long been considered to be a linear function of applied pressure. However, as we have demonstrated in this study, this generally applies only to nonconducting substrates. If substrates are electron-conducting (though still ideally polarizable, no electrode reactions), the linear behavior occurs only at sufficiently low dimensionless pressure differences directly proportional to hydrostatic pressure difference and inversely proportional to solution conductivity. At larger dimensionless pressure differences, the dependence becomes pronouncedly sublinear while the dependence on the coordinate along the flow direction is superlinear (exponential). The extent of nonlinearity also depends on the mechanism of surface-charge formation, charge regulation giving rise to a somewhat less pronounced nonlinearity. Experimental detection of predicted trends calls for the use of rather large applied pressures in systems with relatively large pores in dilute solutions. Alternatively, clear manifestations can be expected in devices where large hydrostatic-pressure differences are induced due to capillarity in water evaporation from nanoporous materials. Experimental data already published for such systems are in good qualitative agreement with the model predictions.